Hilbert schemes, commuting matrices, and hyperkähler geometry

Abstract

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\mathbb{P}}^3\backslash {\mathbb{P}}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive Lie group.

arXiv commentReferences added; presentation improved; minor corrections. To appear in the Journal of the LMS.